Adaptive control using a hybrid-neural model: application to a polymerisation reactor

ADAPTIVE CONTROL USING A HYBRID-NEURAL MODEL: APPLICATION TO A POLYMERISATION REACTOR

F.Cubillos^1*, H.Callejas¹, E.L.Lima² and M.P.Vega²

¹Universidad de Santiago de Chile, Depto. Ing. Química, Casilla 10233, Santiago - Chile.

²Programa Eng. Química, COPPE/UFRJ, C.P.68502, Rio de Janeiro, RJ - Brazil.

¹Depto. Ing. Química, Universidad de Santiago de Chile, Fax : [phone],

Casilla 10233, Santiago - Chile

E-mail: [e-mail]

(Received: March 16, 2000 ; Accepted: January 26, 2001)

Abstract - This work presents the use of a hybrid-neural model for predictive control of a plug flow polymerisation reactor. The hybrid-neural model (HNM) is based on fundamental conservation laws associated with a neural network (NN) used to model the uncertain parameters. By simulations, the performance of this approach was studied for a peroxide-initiated styrene tubular reactor. The HNM was synthesised for a CSTR reactor with a radial basis function neural net (RBFN) used to estimate the reaction rates recursively. The adaptive HNM was incorporated in two model predictive control strategies, a direct synthesis scheme and an optimum steady state scheme. Tests for servo and regulator control showed excellent behaviour following different setpoint variations, and rejecting perturbations. The good generalisation and training capacities of hybrid models, associated with the simplicity and robustness characteristics of the MPC formulations, make an attractive combination for the control of a polymerisation reactor.

Keywords: polymerisation control, neural networks, hybrid-neural model.

INTRODUCTION

The use of neural networks (NN) for modelling dynamic non-linear systems, like polymerisation processes, has been the subject of a great deal of recent research. Some important features of neural network models used for process applications are their capability to act as universal approximators to non-linear functions, their applicability to multivariate modelling and their ability to assess process dynamics (Bishop, 1994). Most of the efforts have been concentrated on developing non-parametric or black box models that correlate a set of input variables with the corresponding outputs.

However, in many cases it seems to be convenient to take advantage of previous knowledge of the non-linear system, frequently expressed as a set of ordinary or partial differential equations representing mass or energy balances. In polymeric processes, the most difficult modelling task remains the determination of some key parameters, such as the reaction rates, and the extensive variation in species properties, which are usually complex functions of the various state variables.

To cope with this problem, hybrid-neural models, which combine prior knowledge included in the phenomenological model with neural networks, have been suggested (Psichogios and Ungar, 1992). These models have proved to be successful for dynamic systems with better generalisation features. In addition, they can be identified with a more reduced set of data than the black-box equivalent model (Psichogios and Ungar, 1992). Thompson and Kramer (1994) classified these hybrid models into two main types: series models with intermediate values (parameters or variables) to be used by the phenomenological model and parallel models compensating for the bias of the parametric model. Van Can et al. (1996) demonstrated that the series approach results in more accurate models with better dimensional and range extrapolation properties than the other approaches.

Given the mentioned advantages, different forms of neural network models have been considered as non-linear models for advanced control algorithms. The ability to approximate almost any non-linear multivariable function, their robustness in a stochastic environment and their simpler construction have encouraged the extensive use of neural networks, both in academic and industrial domains (Ungar et al., 1996).

Specifically, model-based controllers (MBC) with non-linear dynamic models based on neural networks have been studied by Hernández and Arkun (1990), Nahas et al. (1992) and Psichogios and Ungar (1991) with promissory results. Even though they are generally successful, process models based on neural networks need a large number of adjustable parameters (network weights), resulting in two main problems: a big computational load to train the network (weight adaptation) and the possibility of overfitting, with a loss of generalisation capacity when reduced numbers of data pairs are available for training (excess of degrees of freedom).

Those two problems can be minimised if some prior knowledge is used in the construction of the model, such as in the HNM. Since HNM’s neural net has fewer parameters than a pure black box NN model, an on-line training method may be used in order to obtain adaptive HNM. For on-line adaptation, several neural network structures and training algorithms have been informed; one of them, the radial basis function nets (RBFN), has linearly combined outputs which allows the use of RLS-type algorithms for the estimation of the network weight.

In this paper, we used an HNM with an RBFN neural network to control a simulated polymerisation reactor, using two model predictive control schemes. Initially, the bases of hybrid-neural models and RBFN neural nets are presented, followed by a description of the process and HNM synthesis. Finally, the control algorithms, implementation and results are shown and discussed.

HYBRID-NEURAL MODELLING

The hybrid-neural modelling technique uses a process model based on fundamental principles, associated with a neural network (NN), to estimate parameters which are uncertain or difficult to model. In this way, some previous knowledge is incorporated into the black box model to reduce its complexity and improve its adaptation and prediction properties (Psichogios and Ungar, 1992). A schematic representation of this kind of model is shown in Figure 1.

In l992, Psichogios and Ungar applied this hybrid modelling scheme to a simulated bioreactor. They estimated bacterial growth parameters with a one-hidden-layer feedforward neural net. Results indicated an excellent prediction behaviour and good adaptation to noisy data. More recently, Thompson and Kramer (1994) have presented a general framework for model synthesis by combining prior knowledge and neural networks where HNM is classified as a serial semi-parametric approach. HNM has also been proposed as a generalised framework to model particulate drying processes (Cubillos et al., 1996).

Obviously, hybrid models have more information using explicit conservation restrictions than empirical ones, and better results should be expected. Besides, hybrid-neural modelling has two main advantages when compared to purely phenomenological modelling: a better capacity to predict model parameters by using all influencing variables and the possibility of reducing model complexity and absorbing the consequent error during the neural network training procedure (Cubillos et al., 1996). This results in a hybrid model that is generally simpler than the phenomenological one. Hybrid-neural model formulation methodology includes the following steps: 1) development of mass, energy and momentum balances which, together with thermodynamic and transport relationships, allow the link between input and output variables and model parameters; 2) selection of model parameters that will be estimated through NN, and determination of their dependence on input variables; 3) NN structure determination and training, using existing process data.

Radial Basis Function Neural Networks.

Neural networks are non-parametric models formed by process units called nodes or neurons that are ordered in layers and fully or partially interconnected. The main property is their ability to approximate non-linear functions in multidimensional space. Of the many topologies associated with the NN paradigm, the principal type used is the feedforward neural network (FFNN), where information travels exclusively from input to output nodes. For non-linear functional approximations, FFNNs with one hidden layer and sigmoidal-type activation functions are the most popular structures because they have proved to be capable of approximating any continuous non-linear functions (Hornik et al.,1989).

As an alternative, the radial basis function neural networks (RBFN) have been used with success as dynamic models of non-linear processes (Nahas et al., 1992; Chen et al., 1991). Their great attractive is a structure linear in the parameters, which allows that use for some kind of recursive identification algorithm. The RBFN architecture is a two-layer processing structure. The first layer consists of an array of computing units. Each unit contains a parameter vector called the centre, where the confluence is calculated as the euclidean distance between the centre and the network input vector. The unit then passes the results through an activation function. The second layer is essentially a linear combination of the outputs of the hidden layer and its output weights.

The most common activation function used in RBFN is the Gaussian function:

In this work, the goal is to develop a hybrid-neural model, where the unknown parameters (the kinetics rates) are estimated by an RBF neural network that could be trained recursively and included in a predictive-adaptive control scheme.

PROCESS DESCRIPTION AND HYBRID-NEURAL MODELLING

A polymerisation reaction in solutions, carried out in a tubular reactor, has been chosen as a simulated process. The tubular reactor has one feed stream, composed of styrene monomer, benzoyle peroxide as initiator of the reaction and toluene as solvent. The exit stream contains polymer, unreacted monomer, initiator and solvent. Feed flow rate and the reactor jacket temperature can be used as manipulated variables for controlling the quantity and quality of the polymer. In this work the reactor jacket temperature is the manipulated variable and feed flow rate is used as the perturbation variable in the system.

The mechanism of vinyl polymerisation used is given in Table 1. It considers that the lifetime of the radical species is extremely short; therefore, a quasi-steady-state approximation may be assumed.

Based on the previous mechanism, an approximated model of the process was developed with the following assumptions: laminar flow, physical properties variable with temperature, negligible radial and axial dispersion of the species concentration (monomer, initiator, and polymer), parabolic velocity distribution and uniform reactor temperature equal to jacket reactor temperature (Vega, 1995).

According to these assumptions, the material balance for each component in the reactor is given by the following equation:

This model was solved numerically by spatial finite differences, taking 10 segments in the axial direction and 3 discrete segments in the radial direction, which gave a total of 90 differential equations that were integrated using a variable-step RK method. Vega et al,(1997) showed for this process that the dead-time (an important characteristic for control purposes) is more sensitive to the axial than to the radial discretisation, and concluded that a grid containing 10 elements along the axial direction and 3 element along the axial direction was able to reproduce adequately the process dynamics with permissible CPU time for in-line implementation.

The simulation considers a pilot plant tubular reactor with 12 m in length and ¼" ID. Detailed process conditions and parameters may be found in Vega (1995).

Hybrid-neural Model

The synthesis of an HNM for the considered process was carried out according to the considerations outlined above and based on the following assumptions:

(a)The process is carried out in a hypothetical CSTR-type reactor.

(b)The kinetics of each species is unknown.

(c)The net reaction rate of each species (NN_Ri) can be estimated adequately by a RBFN neural network.

Based on these assumptions, mass balance equations for each species are described by:

In order to obtain an explicit predictive model, a simple time discretisation over a sample time Dt in Equation 3 gives the following equation for species concentration at an instant k+1:

NN_Ri indicates the net reaction rates for each species (monomer, initiator, and polymer) and they are estimated by the RBFN as a function of measured process variables such as jacket temperature and exit concentrations at actual and delayed sample times.

The best RBFN topologies were obtained through several off-line tests by randomly varying the jacket temperature and applying the method suggested by Ungar et al. (1996). The data set was split into a training set including 300 patterns and a test set with 200 patterns. For the training, experimental values of the reaction rates were calculated from training set by inverting equation 4. Gaussian activation functions and an orthogonal RLS algorithm were used to obtain the basis centres and output connection weights. The best structure is defined as the neural-net that give the minor deviation among experimental and predicted values over the test set. Taking as base the prediction the monomer reaction rate, a RBFN with three radial bases with both reactor jacket temperature and monomer concentration delayed one sample time as inputs was finally selected. Identical result is obtained by applying the methodology to the others species. Figure 2 gives the complete HNM scheme.

CONTROL ALGORITHMS

The control algorithms used in this work were designed using the non-linear model predictive control (NMPC) framework. In this formulation, an optimisation problem must be solved in each sampling time to find a sequence of M control actions u from the current instant (k) to the instant (k+M-1). The objective function is the difference between the setpoint and the predicted output over the next P time steps. The formulation of this controller is given by (Qin and Badgwell, 1997) as:

In this scheme, the adaptive HNM model gives the prediction of the output plant. Considering the above formulation, two extreme cases were studied:

i)A direct synthesis control scheme (DSC) with M=1, P=1. This means that the manipulated variable, u(k), is calculated such that the output of the process will be equal to the setpoint in the next sampling instant.

(ii) A steady state optimising scheme (SSO), equivalent to using P =, that is to find the value of u(k) such that the process reaches an output steady state equal to the setpoint. This formulation makes use of the phenomenological character of the HNM model, because using equation (3) an explicit expression for the concentration at steady state may be obtained if an appropriate prediction of the net reaction rate is available.

IMPLEMENTATIONS AND RESULTS

Based on the above considerations, a complete simulation shell was implemented using the Simulink dynamic simulation environment including the simulated process, the HNM model with the RBFN trained on-line, and the two NMPC control strategies. In this adaptive control scheme, the centres of the RBFN, obtained from off-line training, are maintained constant, whereas the output weights of the RBFN are re-estimated on-line using an RLS algorithm with a constant forgetting factor.

The variables were suitably scaled between –1 and 1 for RBFN training and for the solution of the optimisation problem. The sampling time was 0.5 min and an SQP routine was used to solve the optimisation problem at each sampling time. Additional bounded limits for the manipulated variable were included. The monomer concentration Cm was selected as controlled variable.

Different tests were carried out considering changes in setpoint and perturbation. Some data were corrupted with white noise. As a comparison, an equivalent DSC controller using a black-box-type neural network model was included. In all these tests it was necessary to carry out an initial sequence of adaptation before operating the system in closed loop. It was observed that the HNM model converges more quickly and it is more stable than the black box model. Figures 3 and 4 show results for setpoint tracking obtained for DSC and SSO controllers, without noise and corrupted with +/-10% of the concentration measurement, respectively. Figure 5 illustrates the capacity of the DSC formulation (HNM and black box) to eliminate a series of perturbations in the feed flow rate.

DISCUSSION AND CONCLUSIONS

Based on the results of this work, it is possible to affirm that the hybrid-neural model is suitable for use in a NMPC-type controller, in an adaptive fashion. Using RBFN to describe the kinetic rates in the HNM, on-line re-estimation was available. Results from different control tests show that the adaptation capacity of HNM is higher than that of a black-box-type neural model, both in the previous initialisation as well as in rejecting perturbations. This is due to the fact that HNM has fewer parameters to estimate than the black box model, and also, the equations of the hybrid model are hard bound, limiting the model outputs and improving the prediction.

Regarding the tests of setpoint tracking (Figure 3), DSC with HNM and the black box model exhibit similar results, with cumulative squares errors (ISE) of 0.11 and 0.12, respectively. Also, it is observed that the SSO formulation followed the setpoint with an asymptotic behaviour, but with worse dynamics (ISE=0.23). The same tendency is observed when the controlled variable is corrupted with white noise (Figure 4), confirming the property of noise cancellation of neural networks.

In general, we can conclude that the formulations studied are adequate for the control of time-varying non-linear systems such as polimerisation processes, where high interference and variations in parameters justify an on-line adaptive adjustment. In particular, the inclusion of an HNM model improves the self-adjustment capacity and prediction notably. It is also possible to affirm that the steady state optimising formulation (SSO) can be used as a suitable model for real time optimisation, with more ambitious indexes such as those used for economic or environmental analyses , but with less dynamic efficiency.

ACKNOWLEDGMENTS

The authors would like to acknowledge grants from the Brazilian Government (CNPq), from the Chilean Government-FONDECYT (grant 1970359 and USACH-DICYT (grant 06-9911CM).

NOMENCLATURE

a_i Output of node i of the RBF layer neural network. c_i Concentration of species i c_io Inlet concentration of species i F Volumetric flowrate I Initiator concentration k_d, k_i, k_IT,
k_Pi k_tr,s, k_tr,m Kinetic parameters yset Setpoint M Control horizon M Monomer concentration n_i Confluence function of node i NN_Ri Net reaction rate of species i estimated by HNM. P Prediction horizon P Polymer concentration R Radical concentration R_i Net reaction rate of species i S Solvent concentration t Time u Manipulated variable V Reactor volume v_z Axial velocity y Controlled variable z Axial position CSTR Continuous Stirred Tank Reactor DSC Direct Synthesis Controller HNM Hybrid-neural Model ISE Integral of the Square Error MBC Model-Based Controller NN Neural Network OSS Optimising Steady State Controller NMPC Non-linear Model Predictive Control RBFN Radial Basis Function Network RK Runge Kutta Integration Method RLS Recursive Least Square SQP Sequential Quadratic Programming

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Publication Dates

• Publication in this collection
  25 May 2001
• Date of issue
  Mar 2001

History

• Received
  16 Mar 2000
• Accepted
  26 Jan 2001